He worked on duality theory for homology while still a student. He went on to lay foundations for the abstract theory of the Fourier transform, now called Pontryagin duality. In topology he posed the basic problem of cobordism theory. This led to the introduction around 1940 of a theory of characteristic classes, now called Pontryagin classes, designed to vanish on a manifold that is a boundary. Moreover, in operator theory there are specific instances of Krein spaces called Pontryagin spaces.

Later in his career he worked in optimal control theory. His maximum principle is fundamental to the modern theory of optimization. He also introduced there the idea of a bang-bang principle, to describe situations where either the maximum 'steer' should be applied to a system, or none.

Pontryagin was a controversial personality. Although he had many Jews among his friends and supported them in his early years, he was accused of anti-Semitism in his mature years. For example he attacked Nathan Jacobson for being a "mediocre scientist" representing the "Zionism movement", while both men were vice-presidents of the International Mathematical Union. He rejected charges in anti-Semitism in an article published in Science in 1979, claiming that he struggled with Zionism, which he considered a form of racism. When a prominent Soviet Jewish mathematician, Grigory Margulis, was selected by the IMU to receive the Fields Medal at the upcoming 1978 ICM, Pontryagin, who was a member of the Executive Committee of the IMU at the time, vigorously objected. Although the IMU stood by its decision to award Margulis the Fields Medal, Margulis was denied a Soviet exit visa by the Soviet authorities and was unable to attend the 1978 ICM in person. Pontryagin also participated in a few notorious political campaigns in the Soviet Union, most notably, in the Luzin affair.

Pontryagin's students include Dmitri Anosov, Vladimir Boltyansky, Mikhail Postnikovand Vladimir Rokhlin.